Let me preface this by saying I have immense respect for E.O. Wilson, the eminent evolutionary biologist and author of the excellent book, Consilience. However, I believe he is totally wrong in the opinions expressed in a recent op ed in The Wall Street Journal, "Great Scientist Doesn't Equal Good At Math' (April 9).
He claims in the piece that for many young people who "aspire to be scientists" the "great bugbear is mathematics". He then poses the rhetorical question: "Without advanced math how can you do serious work in the sciences?"
Whereupon he proceeds to address the question by saying you don't need advanced math, and also fessing up to his own lack of advanced math background. This perceived inadequacy led him to finally sit in (as a tenured professor) on Harvard calculus classes with undergraduates. He concedes he was "only a C student" and "advanced only a small amount".
This experience clearly reinforced his pre-existing perceptions and led him to conclude that "pioneers in science only rarely make discoveries by extracting ideas from pure mathematics." We are asked to accept instead that "real progress" comes by way of "being in the field, writing notes". He does concede, however, that "exceptional mathematical fluency" is required in disciplines such as particle physics, astrophysics and information theory. Indeed! And these were grand discoveries! From the neutron to quarks, to black holes and gravitational deformation of space-time (in general relativity).
And indeed, no one can dispute that Albert Einstein was a great scientist, or Richard Feynman, or Paul Dirac (whose advanced mathematics exposed the positron.) So basically where Wilson takes exception is the biological sciences. He argues:
"The annals of theoretical biology are clogged with mathematical models that either can be safely ignored or, when tested, fail. Possibly no more than 10 percent have any lasting value. Only those solidly linked to knowledge of real living systems, have much chance of being used."
He does say that "if your level of mathematical competence is low plan to raise it" - but "meanwhile know that you can do outstanding scientific work with what you have."
Maybe, but it's interesting that evidently Wilson's own math exposed the deficiencies of one of his biological models allegedly showing support for group selection. Some of the critics, as recently pointed out by Edward Frenkel (Notices of the American Mathematical Society, Vol. 60, No. 7, p. 838) :
"pointed out that one source of error was in Wilson's math."
And of course, evolutionary biology is replete with math that is crucial - but doesn't even rise to calculus level. For example, the success of natural selection is measured by the fitness (w) and the selection value (s): E.g. w = 1 – s. This is generally applied to a species and population after some defined time, say via three genotypes be exhibited in the population: DD, Dd and dd (say the German cockroach species with dominant allele D, denoting resistance to the pesticide dieldrin, and disadvantaged allele d that denotes non-resistance.) Then it can be shown that over successive generations of these roaches the gene frequency (of d) will decrease by:
delta q = -s p q^2 / (1 - sq^2)
where p denotes the frequency of the favored allele, and q the frequency of the "deleterious" allele. A simple table can be constructed (via successive iterations of the previous formula) showing the declining gene frequency of d relative to D.
Needless to say, this sort of math ought to be in EVERY would-be biologist's "wheel house". I mean it doesn't even rise to the level of calculus. The critical insight revealed is that each new (delta q) feeds back to reduce q in the next iteration. Thereby the loss of 'd' genes through selection is balanced by the gain of the 'D' genes that confer reproductive advantage.
Wilson, however, appears to depict any kind of math above basic algebra as some kind of airy-fairy "art form" that really isn't needed as an essential tool. Let's also bear in mind here the critical role of mathematics in the ordering or information, as well as the capacity to represent objective truths clear to everyone. As Frenkel observes (ibid.):
"While our perception of the physical world can be distorted, our perception of the mathematical truths can't be. They are objective, persistent, necessary truths. A mathematical formula means the same thing to anyone, anywhere- no matter what gender, religion or skin color."
For example, the way in which we are able to describe mathematically the twistedness of solar magnetic fields in the vicinity of sunspots, e.g. via the Helmholtz equation, viz.
1/ r [¶/ ¶r ( r ¶/ ¶r)] B + (a)2 B = 0
where r is the radial coordinate, B the magnetic field intensity, and a the "force free parameter". Then the axially symmetric (i.e.- in cylindrical coordinates r, z, q) Bessel function solutions are
B z (r) = Bo Jo(a r)
B z (r) = Bo Jo(a r)
B q (r) = Bo J1(ar)
The biologists, like Wilson, may not need to know this specific illustration, but they ought to be facile with every mathematical tool that can support their own models, or theories! Obviously, they'll never use the advanced mathematics used by those of us who do astrophysics, say the analysis of the evolution of the Sun's largest magnetic fields, but they do need to know enough to, say, amply show any claimed support for group selection.
This gets us to the real issue, which is admirably articulated by Prof. Frenkel (ibid.), i.e. how to improve our math education and to eradicate the fear of mathematics that is rampant (even among science disciplines) and which Wilson gives voice to. Readers already know my stand on this, which is that we need to get more top math specialists into the role of math subject teachers. What we don't want to see is majors in Phys. Ed. or Social Studies teaching subjects like algebra, advanced algebra or god forbid, AP Calculus.
GRE scores show the validity for this view, with education majors typically scoring 1000 total or less while subject specialists score significantly higher. If we are ever to enhance and improve Math Ed in this country we have to change the system!